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Ergodic transformations conjugate to their inverses by involutions

Published online by Cambridge University Press:  19 September 2008

Geoffrey R. Goodson ,
Andrés del Junco ,
Mariusz Lemańczyk  and
Daniel J. Rudolph
Geoffrey R. Goodson
Affiliation:
Department of Mathematics, Towson State University, Towson, MD 21204, USA
Andrés del Junco
Affiliation:
Department of Mathematics, University of Toronto, Canada, M5S1A1
Mariusz Lemańczyk
Affiliation:
Institute of Mathematics, Nicholas Copernicus University, ul. Chopina 12/18 Toruń, Poland
Daniel J. Rudolph
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA
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Abstract

Let T be an ergodic automorphism defined on a standard Borel probability space for which T and T−1 are isomorphic. We investigate the form of the conjugating automorphism. It is well known that if T is ergodic having a discrete spectrum and S is the conjugation between T and T−1, i.e. S satisfies TS = ST−1 then S2 = I the identity automorphism. We show that this result remains true under the weaker assumption that T has a simple spectrum. If T has the weak closure property and is isomorphic to its inverse, it is shown that the conjugation S satisfies S4 = I. Finally, we construct an example to show that the conjugation need not be an involution in this case. The example we construct, in addition to having the weak closure property, is of rank two, rigid and simple for all orders with a singular spectrum of multiplicity equal to two.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 1 , February 1996 , pp. 97 - 124
Copyright
Copyright © Cambridge University Press 1996

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